Method and system for generating aerial imaging flight path

ABSTRACT

A method and system for developing a flight plan for taking images from an area of interest is disclosed. A series of trajectories is determined. Each trajectory is determined based on a logarithmic spiral curve derived from a range of predetermined basis angles and selecting a constant tangent angle between a radial line from the location of an image sensor to a target location, and a tangent line to the logarithmic spiral curve at the location of the image sensor. A set of trajectories from the series of trajectories is selected. The selected trajectories are scaled to cover the area of interest. The selected trajectories are transformed to coordinates corresponding to the area of interest. The set of scaled and transformed trajectories are stored as the flight plan for taking images of the area of interest.

PRIORITY CLAIM

The present application is a continuation of U.S. patent applicationSer. No. 16/053,558 filed Aug. 2, 2018, which claims priority from U.S.Provisional Application No. 62/648,094, filed Mar. 26, 2018. Thecontents of those applications are hereby incorporated by reference intheir entireties.

TECHNICAL FIELD

The present disclosure relates generally to digital image data capture;and more specifically, it is directed toward determining cameraplacement paths derived from logarithmic spiral trajectories forcapturing images.

BACKGROUND

Digital imaging has been a tool used in numerous fields. With recentdevelopments in digital imaging technology, and more recently with theproliferation of aerial drones, digital imaging use is expandingrapidly. Image collection, specifically focused on structure from motion(SfM) processing techniques, is performed widely using platforms such asan aerial drone equipped with consumer grade digital cameras. Animportant aspect of the current state of Structure from Motion (SfM)image collection can be characterized in terms of the impact that cameraplacement trajectories have on the delivered results.

Current UAV data collection strategies for rotor-wing quad-copters, andparticularly those for fixed-wing platforms are largely based on thetried and tested flight plan styles developed for traditional aerialphotography (e.g., orthophotography which are built up from sequentialoverlapping stereo pairs). With traditional orthophotography, the imagedata is acquired using regular parallel linear patterns, withoverlapping images taken along a flight line forming an “image strip.”The parallel overlapping image strips used to survey areas form an“image block.”

Based on current research literature: “High resolution digital elevationmodels are increasingly produced from photographs acquired with consumercameras, both from the ground and from manned aircraft as well asunmanned aerial vehicles (UAVs). However, although such digitalelevation models may achieve centrimetric detail, they can also displaysystematic broad-scale error that restricts their wider use. Sucherrors, which in typical UAV data are expressed as a vertical “doming”of the digital elevation model (DEM) surface, result from a combinationof near-parallel imaging directions and inaccurate correction of radiallens distortion. Using simulations of multi-image networks withnear-parallel viewing directions, enabling camera self-calibration aspart of the bundle adjustment process inherently leads to erroneousradial distortion estimates and associated digital elevation modelerror. This effect is relevant whether a traditional photogrammetric ornewer structure-from-motion (SfM) approach is used; but errors areexpected to be more pronounced in SfM-based digital elevation models,for which the use of control and check point measurements are typicallymore limited.”

Again, from current literature: “Specifically, a regular (oftenrectangular and evenly spaced) pattern of traditional linear, straight,parallel trajectories covering an area, corridor, or point of interestis used when capturing the necessary photographs. In particular, thetraditional, regular, linear, parallel flight lines (used to collect UAVimagery in the recent times) contribute particularly to thewell-established and studied systematic (doming) errors in the 3D pointclouds derived from the SfM processing workflow—even when alternateflight lines are flown in opposing directions. Such flight paths forcollecting images result in the production and propagation of systematicerrors within the SfM workflow processes. These errors are termed the“dome” effect. This effect occurs even though emphasis has been placedon maintaining 60% to 90% axial (forward) overlap and 20% to 40% lateral(side-to-side) overlap for placement of images on-the-ground within thetarget area.”

A generally accepted consensus among researchers is that “convergentimage capture along gently curved trajectories (e.g., circles) clearlymitigates SfM surface deformation in self-calibrating multi-imageblocks.” However, such circular trajectories are neither efficient noreffective enough to cover every potential area or shape.

The traditional (straight-linear-parallel) trajectory scenario (alogical carryover from the longstanding traditional aerial photographyindustry) is generally applied for all types of SfM projects—for pointand linear targets (landmark features), agricultural (rural and urbanareas), and corridors (highways, pipelines, electric power transmissioncorridors, railroads, waterways, etc.). An exception to this heuristicholdover practice might relate to a) cityscape areas with tall buildingsor b) isolated tall towers where circular trajectories might be used. Ofcourse, emphasis is always placed on capturing all available data in theshortest possible time frame—leaving the SfM software/workflow with thebulk of the production (problem solving) effort, particularly in theinstance of using either manned or unmanned aircraft to collect thephotographs where duration of flight is a major concern.

FIGS. 1A-1E depict specific, recent traditional, straight, parallelflight line patterns for unmanned aerial vehicles (UAVs). The processedresults from such obtained images suffer from doming effects/errors.FIG. 1A shows traditional, longer, straight, parallel UAV flight linesfor collecting digital images (digital aerial photographs) over avineyard for the purpose of constructing 3D rendered point cloud modelsof the scene along with the associated orthophotography and DEM usingSfM processing techniques.

FIG. 1B shows traditional, longer, straight, parallel UAV flight linesfor collecting digital images (digital aerial photographs) over acultivated agricultural land for the purpose of constructing 3D renderedpoint cloud models of the scene along with the associatedorthophotography and DEM using SfM processing techniques.

FIG. 1C shows traditional, longer, straight, parallel UAV flight linesfor collecting digital images (digital aerial photographs) overdeveloped rural and/or urban land areas for the purpose of constructing3D rendered point cloud models of the scene along with the associatedorthophotography and DEM using SfM processing techniques.

FIG. 1D shows traditional, longer, straight, parallel UAV flight linesfor collecting digital images (digital aerial photographs) overpopulated urban land areas for the purpose of constructing 3D renderedpoint cloud models of the scene along with the associatedorthophotography and DEM using SfM processing techniques.

FIG. 1E shows traditional, longer, straight, parallel UAV flight linesfor collecting digital images (digital aerial photographs) over ruraland/or urban corridors for the purpose of constructing 3D rendered pointcloud models of the scene along with the associated orthophotography andDEM using SfM processing techniques.

Thus, current methods of deploying cameras (on the ground or in the air)for capturing digital imagery (overlapping digital photographs) for theconstruction of SfM 3D topographical models and associatedorthophotography generally use more traditional, straight, linear,parallel trajectories as shown in FIGS. 1A-1E. It has been demonstratedrepeatedly that such trajectories have a strong inherent tendency tointroduce systematic error (doming effect) into the delivered SfMproducts—often rendering the delivered products less than optimallysuited for the intended purpose.

Thus, there is a need for a method to produce trajectories for imagegathering that avoids the doming deformation effect. There is a need fora system that allows the generation of flight paths to maximizeefficient errorless gathering of images over areas and corridors.

SUMMARY

One disclosed example is a method of developing a path for taking imagesfrom a geographic area of interest. A series of trajectories isdetermined. Each trajectory is determined based on a logarithmic spiralcurve derived from a predetermined range of basis angles and selecting aconstant tangent angle between a radial line from the location of animage sensor to a target location, and a tangent line to the logarithmicspiral curve at the location of the image sensor. A set of trajectoriesis selected from the series of trajectories and the trajectories arescaled to cover the area of interest. The selected trajectories aretransformed to coordinates corresponding to the geographic area ofinterest. The set of scaled trajectories are stored as the path fortaking images.

Another example is a system to determine a path for taking images of anarea of interest. The system includes a storage device storingcoordinates of the area of interest. A trajectory module is coupled tothe storage device. The trajectory module is operative to determine aseries of trajectories. Each trajectory is determined based on alogarithmic spiral curve derived from a predetermined range of basisangles and selecting a constant tangent angle between a radial line fromthe location of an image sensor to a target location, and a tangent lineto the logarithmic spiral curve at the location of the image sensor. Thetrajectory module stores the series of trajectories in the storagedevice. The trajectory module allows the selection of a set oftrajectories from the series of trajectories and scales the selected setof trajectories to cover the area of interest. The trajectory moduletransforms the scaled selected set of trajectories to coordinates tocover the area of the interest. A display module is coupled to thetrajectory module. The display module displays the selected set oftrajectories over an image of the area of interest.

Another example is a system for imaging an area of interest. The systemincludes an image sensor operable to capture images and a flightcontroller having a flight plan. The flight plan is generated bydetermining a series of trajectories. Each trajectory is determinedbased on a logarithmic spiral curve derived from a predetermined rangeof basis angles and selecting a constant tangent angle between a radialline from the location of an image sensor to a target location, and atangent line to the logarithmic spiral curve at the location of theimage sensor. A set of trajectories is selected from the series oftrajectories. The selected set of trajectories is scaled to cover thearea of interest. The selected set of scaled trajectories is transformedto coordinates corresponding to the area of interest. A set of waypointsis generated for the flight plan. An aircraft having the image sensorand controlled by the flight controller in accordance to the flight planflies over the area of interest collecting images.

The disclosed system allows the reuse of a previously constructed denseset of logarithmic-spiral template trajectories constructed on a unitscale basis. The dense set of template trajectories may re-scaled up ordown to reflect a target area's true geometrical size and shape. Thegeometrically re-scaled set of template trajectories are transformed(translated and rotated) into a convenient Cartesian geographicalcoordinate system necessary to frame the trajectories in a target's realworld location and orientation. A number of “favored” trajectories areselected (extracted) from the larger set of (scaled and transformed)template trajectories to provide favorable camera placement locationswhich are easier to access and occupy (fly) to obtain improved SfMresults. Waypoints associated with the selected, favored trajectoriesare downloaded to commercially available mapping software and/or UAVmission planning and/or flight control software to fly the mission(trajectories) and collect the photographs or images for SfM processing.

The above summary is not intended to represent each embodiment or everyaspect of the present disclosure. Rather, the foregoing summary merelyprovides an example of some of the novel aspects and features set forthherein. The above features and advantages, and other features andadvantages of the present disclosure, will be readily apparent from thefollowing detailed description of representative embodiments and modesfor carrying out the present invention, when taken in connection withthe accompanying drawings and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosure will be better understood from the following descriptionof exemplary embodiments together with reference to the accompanyingdrawings, in which:

FIGS. 1A-1E are prior art linear flight patterns for imaging ofdifferent terrains that may result in doming errors;

FIG. 2A is a diagram of the principle of using the natural flighttrajectory of a bird of prey to provide a trajectory with a constanttangent angle focused on the target object;

FIG. 2B is a graph of a logarithmic spiral curve rotated about thex-axis to offer a full convergent view of a target area using a singleflight circuit;

FIG. 3 is a graph of unit-scale logarithmic curves created as a basisfor generating a set of unit-scale flight plan templates;

FIG. 4A-4D are diagrams of logarithmic spiral trajectories using a fullrange of Theta (basis angle) values along with a representative numberof Psi (constant tangent angle) values, while FIG. 4B-4D illustrate thatthe logarithmic spiral trajectories of FIG. 4A are successively rotatedabout the x-axis and mirrored relative to the y-axis direction;

FIG. 5A is an unscaled (unit-scale) composite diagram of the available(derived rotated, and mirrored) logarithmic spiral trajectory templatesfrom FIGS. 4A-4D;

FIG. 5B is a composite diagram of the scaled logarithmic spiraltrajectories from FIGS. 4A-4D covering a large square or rectangulararea;

FIG. 5C is a composite diagram of the scaled logarithmic spiraltrajectories from FIGS. 4A-4D for covering a long narrow corridor;

FIG. 6 is an image of an area of interest that is a long narrowcorridor;

FIG. 7A shows selected scaled trajectories linked to form the basis fora flight plan covering a long narrow corridor;

FIG. 7B shows the selected trajectories from FIG. 5C after beingtranslated and rotated into their true geographic location andorientation and after being linked to form complete flight paths;

FIG. 7C is an image of a long narrow corridor with the selectedtrajectories in FIG. 7B projected on the corridor;

FIG. 7D is a close up view of the selected trajectories in FIG. 7Cshowing waypoints along the flight paths (loops);

FIG. 8 is a diagram of an example drone/UAV aerial imaging system thatincorporates the generated flight patterns;

FIG. 9 is an example flight plan generated by the logarithmic spiral toprovide imaging over a large area of interest, a typical municipalairport;

FIG. 10A is an image of a pilot navigation system (PNAV) display showingan example overview mission plan generated with the disclosed methodsfor the flight plan in FIG. 9 ;

FIG. 10B is an image of the PNAV display in FIG. 10A after entering aflight pattern occupying one of the selected trajectories;

FIG. 10C is an image of the PNAV display in FIG. 10A with an exampleaircraft having flown in approximately to the midpoint of the firstflight trajectory in FIG. 9 ;

FIG. 10D is an image of the PNAV display in FIG. 10A with an exampleaircraft approaching the end of the first flight trajectory in FIG. 9 ;

FIG. 11 is a flow diagram of the process of generating flight paths fromthe previously constructed logarithmic spiral trajectory templates; and

FIGS. 12-13 are block diagrams of example computing devices that may beused for construction of trajectory templates and the conversion of suchtemplates into flight plans.

The present disclosure is susceptible to various modifications andalternative forms. Some representative embodiments have been shown byway of example in the drawings and will be described in detail herein.It should be understood, however, that the invention is not intended tobe limited to the particular forms disclosed. Rather, the disclosure isto cover all modifications, equivalents, and alternatives falling withinthe spirit and scope of the invention as defined by the appended claims.

DETAILED DESCRIPTION OF THE ILLUSTRATED EMBODIMENTS

The present inventions can be embodied in many different forms.Representative embodiments are shown in the drawings, and will herein bedescribed in detail. The present disclosure is an example orillustration of the principles of the present disclosure, and is notintended to limit the broad aspects of the disclosure to the embodimentsillustrated. To that extent, elements and limitations that aredisclosed, for example, in the Abstract, Summary, and DetailedDescription sections, but not explicitly set forth in the claims, shouldnot be incorporated into the claims, singly or collectively, byimplication, inference, or otherwise. For purposes of the presentdetailed description, unless specifically disclaimed, the singularincludes the plural and vice versa; and the word “including” means“including without limitation.” Moreover, words of approximation, suchas “about,” “almost,” “substantially,” “approximately,” and the like,can be used herein to mean “at,” “near,” or “nearly at,” or “within 3-5%of,” or “within acceptable manufacturing tolerances,” or any logicalcombination thereof, for example.

The present disclosure relates to a method and system for prescribingand constructing nontraditional camera placement trajectories (each aseries of specific waypoints) to more effectively capture digital imagenetworks for the production of geometrically and geographically correct3D point cloud models using Structure-from-Motion (SfM) techniques. Thespecific waypoints along gently curving trajectories may guidephotographers in the placement of cameras. The specific waypoints mayalso provide pathways for ground vehicles and/or flight lines formanned/unmanned aircraft and/or trajectories for water-borne craftcarrying an adequate payload of vertical and/or oblique cameras ofvarious types sufficient for the prescribed imaging of the point, area,or corridor of interest. The unique, gently curving, convergent,non-parallel, continuously changing shape (characteristics) of cameratrajectories produced using the disclosed method and system arenecessary and sufficient to significantly contribute to the resolutionof identified systematic errors previously experienced in SfM productionof 3D topographic models from digital photographs captured usingtraditional straight, linear, parallel camera trajectories.

The present disclosure, invoking a deliberately re-directedimplementation of well-understood and observed natural phenomenon alongwith solid mathematical principles, embodies traits and properties whichare consciously directed at countering the observed shortfallsencountered in the current and popular traditional digital image capturetechniques supporting the SfM workflow—the mitigation of “doming”systematic errors. That is, the present disclosure exposes new method(s)for addressing the systemic doming problem prevalent in some (if notmost) traditional SfM data modeling endeavors.

Initially, the disclosed camera trajectories, which mimic spiral shapesobserved in nature including the “attack” flight paths of large birds ofprey (falcons and eagles), are constructed on a “unit” scale in thepolar (r, Theta) coordinate system as modeled using the logarithmicspiral curve equation which follows, in which r is the position vector(from the curve's origin to the viewing point), Psi is the constanttangent angle relative to the curved path at position r, and Theta isthe basis angle of the defining equation. Such a curve 200 is shown inFIG. 2A. FIG. 2A also illustrates a tangent line 202, a radial line 204and an origin point 206. The origin point 206 represents the object orprey of interest. The angle between the radial line 204 and thehorizontal of the origin point 206 (Theta) varies continually as thetrajectory is traversed. The angle between the radial line 204 and thetangent line 202 (Psi) remains constant as the trajectory is traversed.The intersection of the radial line 202 and the tangent line 204represents a current occupied point 208 of the trajectory, which is thecurrent location of the camera (bird's eye).

FIG. 2A illustrates a very important characteristic, the constanttangent angle, of the logarithmic spiral trajectory that large birds ofprey take advantage of. The constant tangent angle (Psi) is formedbetween the radial line 204 from the origin (prey) location to thecamera location (falcon's eye) and the tangent line 202 to thelogarithmic spiral curve 200 at the camera's current location (falcon'seye). This tangent angle remains constant along the entire length of thelogarithmic spiral curve 200. Thus, with its head held stationary anddirected straight ahead, the falcon's eye (camera) is constantly focusedon the prey (origin) throughout the entire traverse of the logarithmicspiral curve (flight path). In the falcon's case, this characteristic isa direct result of the construction of the falcon's eye; but it is acharacteristic that can be modeled (and taken advantage of) by cameraoperators on both the ground and in the air to reduce image modelingerror when SfM processing techniques are employed. The errors are thusreduced by collecting additional convergent images taken at an obliqueangle (constant tangent angle) to the trajectory of a primary(vertical/oblique) camera or cameras. The image modeling error is thusreduced even when an additional convergent (constant tangent angle)oblique camera is not used, since the orientation of the images taken bythe primary camera or cameras is constantly changing as the primarycamera(s) is moved along the prescribed trajectories.

FIG. 2B illustrates a logarithmic spiral curve 250 which can be foundthroughout nature. Described mathematically, the curve is of the generalform shown in FIG. 2B. The logarithmic spiral curve is generated fromthe following equation:r=+e{circumflex over ( )}((Theta−π)*cot(Psi))  (1)

In this equation, r is the distance from the origin (target) to thelogarithmic curve, Theta is the predetermined basis angle, and Psi isthe constant tangent angle. In this general equation, the primaryvariables are constrained as follows: (0≤Theta≤2π) and (0≤Psi≤π) withπ=3.1416 and e (Euler's constant)=2.71828. Alternatively, thelogarithmic spiral curve may be approximated using Fibonacci sequencenumbers as well as other formulations involving a mathematical series ofnumbers. For example, construction of the trajectories usinglogarithmic-spiral equations or alternative approximation techniquessuch as Fibonacci numbers, mathematical series, Bezier curves, orB-splines, or curve-fitting techniques may be used.

In the disclosed method, the general form of the logarithmic spiralequation is constrained in order to provide a more specialized andfundamental result. As (r=e{circumflex over ( )}((Theta−π)*cot(Psi))produces results in polar coordinates (r, Theta), the polar coordinatesare converted to two-dimensional Cartesian coordinates (x,y) for displayin FIG. 2B by computing (x=+r*cos(Theta)) and (y=+r*sin(Theta)). Anupper half curve 250 of FIG. 2B illustrates the exponential functionwith (0≤Theta≤π) and (Psi=(π/6)). The curve 250 is situated above thex-axis with a true unit scale (−1≤x≤+1), where (0≤Theta≤π) and(Psi=(π/6)). Thus, a value is assigned to Psi (the constant tangentangle); while Theta (basis angle) is cycled through its entire range ofvalues to construct a particular logarithmic spiral curve. Thesecomputed results are shown by the curve 250. This procedure (with adifferent value of Psi) may be used to construct other logarithmicspiral curves. For practical applications (practical scaling purposes),the logarithmic curves are considered to exist in a range (−1≤×≤0) alongthe negative x-axis. However, the extension of the logarithmic curvesbeyond the y-axis (x>0) naturally facilitates a convenient transitionfrom one trajectory such as the curve 250 to another trajectory based ona lower half curve 252 when the final flight plan is constructed.

The lower half curve 252 of FIG. 2B illustrates the rotation of theexponential function about the horizontal x-axis as (x=−r*cos(Theta))and (y=+r*sin(Theta)). The upper and lower logarithmic spiral curves 250and 252 of FIG. 2B are two separate trajectories intended to divide theregion into two equal portions divided by the x-axis. When the twoseparate trajectories 250 and 252 are traversed in opposing directions,one after the other, a complete circuit (loop) of a central target (areaof interest) 254 can be circumscribed. This circuit may be used inproviding a flight plan.

FIG. 3 shows constructed logarithmic spiral curves with a constant anglebetween the position vector and the tangent of the spiral trajectory,specified for each individual curve. FIG. 3 shows the general shape ofthe logarithmic spiral curve defined by the exponential function(r={circumflex over ( )}(Theta−π)*cot(Psi)) where (0≤Theta≤π), (Psi=30,45, 60, and 90 degrees), and (π=3.1416) with (e=2.71828). For (Psi=90degrees), the logarithmic spiral curve becomes a circle if Theta isallowed to vary between zero and 2π, which is a special case of thelogarithmic spiral curve formulation. The constructed curves include acurve 300 at Psi=30 degrees, a curve 302 at Psi=45 degrees, a curve 304at Psi=60 degrees and a curve 306 at Psi=90 degrees. The curves can beconstructed for any value of Psi, varying Psi in equal or unequalincremental interval values. As the value of the constant tangent angle(Psi) is changed, the eccentricity of each resulting curve differs fromthat of the other curves, while the maximum distance between the curvesis reasonably consistent and varies (increases with increasing values ofPsi) at a reasonably consistent and predictable rate.

FIG. 3 depicts the “unit scale” of the chosen formulation of thelogarithmic spiral curve. That is, for values of Psi less than or equalto 90 degrees, the logarithmic curves exist in a range on: a) the x-axisas [−1≤x≤+1]; and b) the y-axis as [−1≤y≤+1]. This unit scale of thelogarithmic spiral curve is convenient. This means that the initiallyconstructed curves are reusable. The initial unit scale curves can bere-scaled to represent (reflect) the geometrically correct size andshape of the intended target area. Additionally, the unit scalelogarithmic spiral curves may be rotated about the x-axis and mirroredrelative to the y-axis direction. The rotation and mirroring operationsare performed regardless of the intended target area, whether the targetarea exists as a regular shape (e.g., point, triangle, square, circle,etc.), an irregular shape (e.g., rectangular, oval, odd-shape, etc.), ora long narrow segmented corridor area.

FIG. 4A illustrates a particular example family of logarithmic spiraltrajectories with (0≤Theta≤π) and (Psi=π/36, π/18, π/12, π/9, π/7.2, andπ/6). These chosen values of the Theta and Psi angles offer ratherequally spaced set of trajectories (at their approximate midpoint ormaximum y value). Additional values of Psi could be used to add moretrajectories to those shown in FIG. 4A. The trajectories are non-linear,non-parallel, gently curving, convergent, and constantly changingdirection throughout the entire trajectory. Additionally, thetrajectories exist on a unit scale in both the x-axis and y-axisdirections. At a value of Psi=30 degrees, the unit length ratio (MaxX/Max Y) of the logarithmic spiral trajectories is approximately((x/y)=(5/1)), which is an indication of the eccentricity of thisparticular trajectory. In FIG. 4A, the x-coordinates are computed as(x=+r*cos(Theta)); and the y coordinates are computed as(y=+r*sin(Theta)).

FIG. 4B presents the trajectories shown in FIG. 4A rotated about thehorizontal x-axis. That is, the logarithmic spiral trajectories arefirst computed as (r=−e{circumflex over ( )}((Theta−π)*cot(Psi)) with(0≤Theta≤π) and (Psi=π/36, π/18, π/12, π/9, π/7.2, and π/6); thex-coordinates are computed as (x=−r*cos(Theta)); and the y coordinatesare computed as (y=+r*sin(Theta)). The trajectories of FIG. 4B providethe first complement to the trajectories of FIG. 4A.

FIG. 4C shows a mirror image of the trajectories shown in FIG. 4A. FIG.4C presents the trajectories shown in FIG. 4A mirrored with respect tothe vertical y-axis. The logarithmic spiral trajectories in FIG. 4C arecomputed as (r=+e{circumflex over ( )}((Theta−π)*cot(Psi)) with(0≤Theta≤π) and (Psi=π/36, π/18, π/12, π/9, π/7.2, and π/6); thex-coordinates are computed as (x=(−r*cos(Theta))−1); and the ycoordinates are computed as (y=+r*sin(Theta)). The trajectories of FIG.4C provide the second complement to the trajectories of FIG. 4A.

FIG. 4D shows the trajectories shown in FIG. 4A first mirrored (as shownin FIG. 4C) and then rotated about the horizontal x-axis. FIG. 4Dpresents the trajectories shown in FIG. 4C rotated about the horizontalx-axis. The logarithmic spiral trajectories in FIG. 4D are computed as(r=−e{circumflex over ( )}((Theta−π)*cot(Psi)) with (0≤Theta≤π) and(Psi=π/36, π/18, π/12, π/9, π/7.2, and π/6); the x-coordinates arecomputed as (x=((+r*cos(Theta))−1); and the y coordinates are computedas (y=+r*sin(Theta)). The trajectories of FIG. 4D provide the thirdcomplement to the trajectories of FIG. 4A.

Once constructed, the unit-scale trajectories in FIG. 4A are rotatedabout the x-axis and mirrored with respect to the y-axis to become a set(family) of reusable trajectories. Next, a few trajectories can beselected (based on their suitability [eccentricity] to cover a targetarea of interest) to be re-scaled and transformed (translated androtated) into a real-world Cartesian coordinate system. This providesoptimal camera placement trajectories necessary and sufficient tosupport the collection of digital images (photographs) more suitable tobe processed using SfM techniques. The example method and system ofconstructing trajectories for aerial/terrestrial collection of verticaland/or oblique digital photographs provides a unique diversity of viewaspects for all types of targets. The technique affords a constanttangent (oblique) camera mounting angle for point and area type targets,thus avoiding the need to use gimbal-mounted cameras with theirassociated control issues. Also, the logarithmic spiral trajectorytempers the propensity of SfM techniques to produce results exhibitingthe “dome” effect error. Thus, the method tempers the propensity forintroducing the observed systematic error (doming effect) when the SfMworkflow is subjected to digital imagery captured using traditionalparallel and linear camera trajectories.

The process of determining a flight trajectory to reduce image modelingerrors includes the following procedure. First, a dense set of reusablelogarithmic-spiral trajectory templates is constructed on a unit scalebasis. The dense set of trajectory templates is re-scaled (up or down)to reflect a true geometrical scale of the target area's size and shape.The geometrically re-scaled set of trajectory templates are transformed(translated and rotated) into the Cartesian geographical coordinatesystem necessary to frame the trajectories in the real world locationand orientation of the target area. A number of “favored” trajectoriesare selected (extracted) from the larger set of (scaled and transformed)trajectory templates to provide favorable camera placement locationswhich are easy to access and occupy to obtain improved SfM results.Waypoints associated with the selected, favored trajectories aredownloaded to commercially available mapping software and/or UAV missionplanning software to fly the mission and collect the photographs for SfMprocessing.

The resulting favored trajectories may be selected by a camera operatorbased on the physical and operational characteristics of the chosencamera (image size on target), the axial/lateral image overlaprequirements of the SfM software, the forward speed of movement of thedeployment mechanism, and the camera(s) mounting angle. Other physicaland operational characteristics (considerations) of the chosen cameramay include the sensor array size, lens focal length, shutter speed, andimage size at the chosen range to the target. The axial/lateral imageoverlap may include forward or side-to-side image overlap requirementsof the SfM software for processing the collected images. The cameramounting angles are considered if the camera is moving along a selectedtrajectory. For the moving camera, the selected trajectories provide aninherent operational efficiency with one occupied trajectory practicallyaligning with the next (to-be-occupied) trajectory. This trajectoryalignment characteristic provides a continuous path and thus savestransition time and energy for a camera platform such as a drone or apiloted vehicle.

The above process, characteristics, and considerations may be seen inFIGS. 5A-5C. FIG. 5A shows all the trajectories in FIGS. 4A-4D as acomposite set of trajectories covering a well-defined area and allexhibiting the characteristics of being gently curving, convergent withrespect to the horizontal x-axis (target area), and continually changingdirection throughout the covered area, while: a) maintaining aunit-scale along the horizontal x-axis and the vertical y-axis; b)offering a known constant tangent angle for each individual trajectory;and c) possessing the ability to be scaled (adjusted) appropriately(e.g., conform to the size and shape of the target area).

The trajectories in FIG. 5A exhibit the characteristics of beingnon-traditional (neither linear nor parallel, gently curving, convergentwith respect to the x-axis, and constantly changing direction) incontrast to traditional (straight, linear, parallel) trajectoriescurrently being used as aircraft flight trajectories for aerialphotography. The composite set (family) of logarithmic spiraltrajectories in FIG. 5A forms the basis from which to create practicalcamera trajectories suitable for photographing target areas of varioussizes and shapes. The set of trajectories in FIG. 5A may be used as acomprehensive set of unit-scale logarithmic-spiral trajectory templates.This set of templates may be reused over and over as a basic set and thetemplates may be scaled to fit any target environment.

FIG. 5B shows the set of trajectory templates shown in FIG. 5A scaled tocover a large target area 500 of (10,000×10,000) feet or meters for asquare area or a large target area 500 of (10,000×20,000) feet or metersfor a square area, depending on which trajectories are selected foreither application. Of course, any preferred unit of measurement may beused for the scaling process. In this example, the scale factor in thex-direction is set to (X-Scale=10,625) while the scale factor in they-direction is set to (Y-Scale=50,000). As explained above in referenceto FIG. 4A, the unit length ratio of the logarithmic spiral trajectory(Psi=30 degrees) is approximately ((x/y)=(5/1)), a convenienteccentricity for a regular (square or rectangular) target area.

Setting the scale factor in the x-direction to 10,625 extends thetrajectories beyond the target length (10,000′) by 625′ to ensure amplecoverage of the full length of the target area. The trajectory shapes onthe left/right ends of FIG. 5B are maintained; and the more-exaggeratedtrajectories (larger values of Psi) do extend to the right (x>0) beyondthe origin, facilitating transition from one trajectory to another.

The unit scale ratio of the logarithmic spiral trajectory (Psi=30degrees) is approximately ((x/y)=(5/1)) as explained in reference toFIG. 4A, the y-scale factor is set (YScale=(10,000*5=50,000) to ensurecoverage up to and including the y-extent (20,000′) of an intendedtarget area 500 in the central portion of the area. This setting (scale)leaves the extreme corners unexposed (uncovered). In order to cover theextreme corner areas, either the value of the y-scale factor may beincreased, or additional template trajectories may be added using largervalues of Psi in increments up to and including the value (Psi=π/2),which represents a circle trajectory. This scaling process may beeffectively implemented using either a commercially availablespreadsheet software product or it may be programmed via a customsoftware package.

FIG. 5C presents the set (family) of trajectory templates shown in FIG.5A scaled to cover a target area 510 that is a long narrow corridor of(10,000×800) feet or meters. The intended target area 510 begins at theorigin (0, 0) and terminates at about 10,000 feet in the “−x” direction.The trajectories are extended beyond the left intended end point(x=−10,000′) by setting the x-scale factor to a value of 10,625 toillustrate a desire to ensure more-than-adequate coverage of the leftend point. The trajectories naturally extend beyond the right (intended)end point of the target area (x>0) to illustrate an opportunity to takeadvantage of the fact that certain trajectories naturally extend beyondthe origin such that the right end of the target area will be coveredfully as a matter of course.

The scale factor in the y-direction is set to (YScale=10,000/5=2000) toensure generous coverage of the central portion of an 800′ widecorridor. Varying degrees of eccentricity are exhibited between thevarious trajectories. Again, a small portion of the extreme corner areasof the long narrow rectangular corridor will not be covered. However, inorder to cover the corner areas, either the value of the y-scale factormay be increased or additional trajectories may be selected using largervalues of Psi in increments up to an including the value (Psi=π/2),which represents a circle trajectory. FIGS. 5B and 5C taken together,illustrate the versatility and flexibility of the logarithmic spiraltrajectory templates when using: a) the scaling operation; and b) theeccentricity characteristic (constant tangent angle) to cover two verydifferent target areas/shapes.

FIG. 6 shows an example of a bounded long narrow corridor shown in aphotographic image 600 or an area of interest. The image 600 contains along narrow, legally surveyed/bounded corridor 300′ wide, which is anexisting electric power transmission right-of-way located insoutheastern Alabama, as displayed in Google Earth. Typically for aerialphotography purposes, the corridor of interest would be 100′ wider (50′wider on each side of the centerline) than the legal corridor. As such,the image 600 is geographically correct referencing WGS84(Latitude/Longitude) coordinates. The actual corridor of interest (asevidenced by a centerline 602) is shown bounded by a buffer zone 610.Thus, the corridor to be photographed, in this example, is approximately10,000′ long and 400′ wide and includes a left end 612 and a right end614. The target segment is bounded by (in between) two turn points, astructure location 616 on the right end and a structure location 618 onthe left end. Additional corridor lengths (segments) are shown on eachend 612 (left) and 614 (right) of the target corridor segment ofinterest 610 for reference and orientation purposes. The transmissioncorridor itself terminates into an electric power transmissionsubstation 620 shown in the upper left of the figure and it terminatesinto what will become a next corridor segment of interest 622 shown inthe lower right portion of the image 600.

Long narrow corridors of this type are constructed in longerstraight-line segments. For imaging purposes, each segment would beflown geographically in sequential order. For example, in this case,each corridor segment may be flown from left to right (west to east) onesegment after another. In this example, the target segment terminates attransmission tower locations known as “points of intersection” orcorridor turn points such as the structures at the turn points 616 and618. As such, the photographic images collected over the target corridorsegment of interest should include these two structures or transmissiontowers.

In order to overlay the previously constructed logarithmic spiraltrajectories (Cartesian coordinates) onto the geographic area of thelong narrow corridor FIG. 6 , the geographic coordinates(Latitude/Longitude) used to depict the corridor in FIG. 6 first must beconverted into a suitable Cartesian coordinate system (the Alabama-EastState Plane Coordinate System in this example) using appropriate unitsof measure (US Survey FEET). For example, this conversion can beaccomplished using a commercial software package such as Global Mapperand exported as a .shp (shape) file for display purposes.

FIG. 7A shows a few specific (sample) trajectories similar to the set oftrajectories shown in FIG. 5C. However, these trajectories have beenspecifically constructed and scaled to cover a more precise area ofinterest (10,000′×300′), the legal transmission corridor, and, moreimportantly, to illustrate the efficiencies gained when differentcomplementary trajectories, with the same eccentricity (Psi values), arelinked to facilitate transitioning from one trajectory to anotherforming a single circuit (loop) circumscribing a given target area. Thetrajectories of FIG. 7A exist in a Cartesian coordinate system tofacilitate simple mathematical operations such as scaling. Thetrajectories include a trajectory 700, a trajectory 702, a trajectory704, a trajectory 706, a trajectory 710, and a trajectory 712. Four ofthese trajectories (700, 702, 704, and 706) are derived from the Psivalue being 34 degrees. The two other trajectories (710 and 712) arederived from Psi being 19 degrees. The eccentricity of trajectories withPsi values of 34 degrees and 19 degrees are deemed sufficient tosatisfactorily cover the legal corridor area (size and shape) ofinterest in this example. In this example the trajectory 712 is notinitially intended as an active flight path, but it is available shouldit be needed to ensure complete coverage of the central target area. Thefive (5) trajectories 700, 702, 704, 706, and 710 were chosen, in part,because they cover an area of (10,000′×300′), matching the size andshape of the legal corridor of interest (between the structures at turnpoints 616 and 618) in the area of interest 600 shown in FIG. 6 andthese five trajectories can be used to traverse (loop around, startingon a left end point 718) the entire length of the corridor of interesttwice (transitioning between two trajectories per loop) while leavingthe fifth trajectory 710 to reach (moving left to right) a “next”segment of interest point 716. A first loop is formed by: a) enteringtrajectory 700 from the left end point 718; b) traversing trajectory 700left-to-right' c) smoothly transitioning onto trajectory 706 at theright end point 716; and d) traversing trajectory 706 right-to-left tothe left end point 718. A second loop is formed using trajectories 702and 704 by: a) smoothly transitioning (at the left end point 718) ontothe trajectory 704 from the trajectory 706 of the first loop whilemoving right-to-left; b) traversing trajectory 704 left to right; c)smoothly transitioning to the trajectory 702 at the right end point 716;and d) traversing trajectory 702 right-to-left until the left end point718 is reached. These particular trajectories are selected because, ifan aircraft were to enter onto the first loop (at the left end point718) created from the first trajectory 700 above the x-axis and followit to the right end (origin), the aircraft would transition smoothlyonto the second trajectory 706 of the first loop, below the x-axis, andfollow it back to the left end or beginning point 718. Moving directlyacross the left end point 718 from the trajectory 706 onto thetrajectory 704 (above the x-axis), the aircraft would transitionsmoothly from the first loop onto the second loop and proceed to theright end (origin) point 716 and proceed to smoothly transition onto thetrajectory 702 of the second loop (below the x-axis) and continue backtoward the left end (x=−10,000′) and the beginning point 718. At thecompletion of the two loops, the aircraft has made two complete circuitsaround and through the area of interest. Ultimately, it is desirable tomove along the long narrow corridor from left to right in order to moveonto and survey a second segment of the corridor of interest, a “next”segment to the right beginning at the end point 716, as shown in FIG.7A. As such, the aircraft makes a smooth and efficient counterclockwiseturn, upon reaching the left end point 718 of the second loop, to occupythe “final pass” trajectory 710 traveling left-to-right in order tooccupy and survey the “next” segment of the long narrow corridorextending to the right beyond the bounds of FIG. 7A. The next segmentwill have its own set of selected trajectories depending on its size(length and width) and shape.

FIG. 7B depicts selected scaled trajectories from FIG. 5C after eachtrajectory has been transformed (mathematically) into a real-worldCartesian coordinate system. Each scaled trajectory has been translatedto a geographic origin at a point 720. In this example, a turn pointorigin 720 represents an “Ahead” structure location at coordinates(Easting=826763.8′ and Northing=270196.8′); and each trajectory has beenrotated (clockwise) about the vertical z-axis of the origin. A left turnpoint 722 represents a “Behind” structure. Thus, the waypoints of eachtrajectory are scaled and transformed into the real-world Alabama-EastState Plane (Cartesian) Coordinate System (NAD83/NAVD88) with horizontalunits of measure being US Survey Feet and vertical units of measurebeing International Feet. The mathematics of the trajectorytransformation process are well known and include, in vector form, amethod for quickly determining the direction cosines required to performthe rotation step of the transformation process. Additionally, theselected, scaled, and transformed complementary trajectories have beenlinked to form flight loops (similar to the approach illustrated in FIG.6 ) sufficient for efficiently covering the area of interest.

The trajectories are placed on-the-ground taking into account the baseelevation (z=270.8′ above sea level) of the Ahead structure 720 and thebase elevation of the Behind structure 722. The elevation of thetrajectories is important because any aerial photography operations(digital image collection) will need to be performed along the selectedtrajectories at a specific height-above-ground sufficient to maintain arequired image footprint size on the ground. The required imagefootprint size is the size necessary and sufficient to produce anaccurate resultant resolution and overlap, leading to an accurate SfMDEM model.

FIG. 7B depicts the scaled and transformed trajectories (the foreground)as an overlay onto physical boundaries 730 of a narrow transmissioncorridor (the background). The corridor's physical boundaries 730isolate the area of interest (imaging target area) in this example. Moreexplicitly, the entire area within the physical boundaries 730 andbetween the Ahead structure at point 720 and the Behind structure atpoint 722 constitutes the imaging target area. As shown in FIG. 7B, thisimaging target area is completely covered by the selected trajectories.In order to produce the image in FIG. 7B, a centerline 732, the physicalboundaries 730, and transformed trajectories 734 were imported into acommercially available software package (Global Mapper) in the form ofCartesian coordinates (Alabama-East State Plane Coordinates,NAD83/NAVD88 with horizontal units of US Survey FEET and vertical unitsof International Feet).

After the transformation of the trajectory waypoints from a localCartesian [x, y] system to a real world Cartesian [X, Y, Z] coordinatesystem, such as the Alabama-East State Plane Coordinate System, areadily available commercial mapping software package, such as GlobalMapper, can be used to input the trajectories and export a kmz or kmlfile (world coordinates [Latitude/Longitude, elevation]) so that thetrajectories and/or waypoints may be displayed using display softwaresuch as Google Earth for visualization and quality assurance purposes.Similarly, waypoints of the selected trajectories in the form of the kmlfile can be loaded into available commercial mission planning software,such as ForeFlight, for a manned aircraft mission plan. Waypoints mayalso be in the form of a .csv file that may loaded into availablecommercial mission planning software such as DJIFlightPlanner or LitchiMission Hub for UAV aircraft/drones.

FIG. 7C shows an image of the area 600 in FIG. 6 with the set ofselected trajectories 734 in FIG. 7B, displayed using Google Earth,plotted over the long narrow corridor of interest and covering theexisting electric power transmission right-of-way. This is the qualityassurance check. The selected trajectories 734 are now in a form that isgeometrically and geographically correct. The selected trajectories 734may be traversed, in the order prescribed in FIG. 7B, to photograph thecorridor of interest by moving any of several cameras (mountedvertically and/or obliquely to the deployment mechanism) along theselected trajectory loops (waypoint to waypoint).

Traversing the trajectory loop(s) in the manner prescribed in FIG. 7B,camera(s) mounted obliquely to the body axis of the deployment mechanismshould be pointed toward the right (convergent) side of the occupiedtrajectories when movement is in a clockwise direction to capturedigital images in a most effective manner. Other camera(s) could bepointed either: a) vertically (nadir), b) obliquely forward or aft inthe vertical plane along the direction of movement [tangent to theoccupied trajectory] to capture images in a more sweeping movementpattern; and/or c) oblique relative to the horizon and in the directionof the constant tangent angle (Psi) to remain focused in the directionof the end point of the trajectory while capturing a larger overview ofthe entire scene. However, even if oblique cameras are not used, theimages captured by vertical (nadir) cameras moving along the logarithmicspiral trajectories will exhibit a diversity of views not possible whentraditional (linear and parallel) flight paths are used.

The complementary logarithmic spiral trajectories (constructed using arotation about the x-axis with a mirroring along the y-axis direction)do cross each other in the central portion of the area of interestproviding a unique diversity of view aspects for cameras mounted in boththe vertical (nadir) and oblique orientations. The logarithmic spiraltrajectories are constantly changing direction over the entire length ofthe trajectory. The resulting overlapping images with diverse viewperspectives (constantly changing image orientations, one image relativeto another) positively impact the results obtained from the SfM process.Thus, systemic error will not persist from single images through full(entire) image blocks as processed using a SfM workflow. The error fromany single image will not be compounded (carried over) from one image toanother through full image blocks.

FIG. 7D shows a “zoom” view of the selected trajectories in FIG. 7C toillustrate (visualize in real world coordinates) unique characteristicsof the example logarithmic spiral trajectories. These characteristicsinclude the constantly changing directions of the trajectories. Thesechanges include, gently curving (non-linear), convergent, (non-parallel,yet consistently focused), and differential eccentricity while beingversatile and flexible in their application and use. Thesecharacteristics provide the SfM workflow with many digital overlappingimages that are slightly and smoothly (incrementally) rotated andtranslated relative to each other while maintaining the required forwardand lateral (side-to-side) overlap. At no point along the corridorstationing (axial distance along the corridor centerline) are thetrajectories straight or parallel to one another.

FIG. 7D shows waypoints 740 that are unevenly distributed throughout thetrajectory 734. The distribution is less densely spaced when lesseccentricity is evident and more tightly spaced when more eccentricityis evident. In this example, the waypoints 740 are shown as black dotsalong one of the trajectories 734 for purposes of illustration. Thewaypoints 740 serve as navigation aids/points for moving along thetrajectories either on the ground such as via a mobile ground vehicle orthrough the air via a manned or unmanned aircraft. Such vehicles may beguided by either manual or automated piloting techniques.

The method described above offers a tool for consistently collectingoblique overview images along the logarithmic spiral trajectories. Thattool is the constant tangent angle Psi shown in FIG. 2A. Takingadvantage of the constant tangent angle requires only that (while movingforward) the camera operator maintain the constant tangent anglebetween: a) the body axis of the mobile camera platform; and b) theline-of-sight to the target from the operator's perspective. Maintainingthis constant tangent angle as the mobile camera platform moves forwardensures that the mobile camera platform is navigating along alogarithmic spiral trajectory. An example for implementation of thisapproach would be to use a modern GPS device or “pilot/operator”navigation device to: a) record a waypoint in the device at the targetpoint location (e.g., end of the collective trajectories), b) maneuverthe mobile camera platform to establish the constant tangent angle asthe observed angle (difference) between the mobile camera platform'sbody axis heading and the device-determined heading to the targetwaypoint, while c) maneuvering the mobile camera platform forward in amanner that maintains that constant difference angle between the mobileplatform's body axis heading and the device-determined heading to thetarget location. In fact, this approach could even be implementedvisually/manually by mounting a physical “sighting” mechanism (any typeof physical marker attached to the body of the mobile camera platformand offset from the body axis at the desired constant tangent angle) forthe camera operator's use to maintain alignment of the “sight” and thetarget (e.g., line of sight).

The moving platform (camera) operator will ensure that the properheight-above-ground is prescribed and maintained throughout the imagecollection operation to acquire images with the appropriate amount offorward and side-to-side overlap. Also, the moving platform (camera)operator will ensure that an appropriate forward speed is prescribed andmaintained to ensure that the images captured are not blurred. Thecamera's prescribed height-above-ground and forward speed will bemaintained as the camera is traversed along any particular logarithmicspiral trajectory.

The desired flight plan includes a set of selected, scaled, andtransformed logarithmic spiral trajectory waypoints. The flight plan maybe used as an input to mission planning software and systems. Suchmission planning software and systems may be used for either unmannedaircraft such as UAVs or in the form of pilot navigation software andsystems for manned aircraft. In addition, such techniques may be usedfor ground or water based vehicles. For example, a plan may be preparedfor an automated or robotic autopilot system for unmanned or mannedground or water based vehicles. Such systems may be specificallydedicated for navigation and guidance. Alternatively, more genericsystems such as GPS mapping devices or GIS mapping devices may be used.Alternatively, generic computing devices such as tablets or smart phonesmay execute GPS or GIS mapping software incorporating the abovementioned path determination techniques.

FIG. 8 is a diagram of an example UAV aerial imaging system that may beused to determine logarithmic trajectories and implement a flight planto image an area of interest. The system allows the determination andexecution of a flight path. The system includes a computing device 800that determines an appropriate flight pattern over an area of interestas explained above with reference to FIGS. 2A-7D.

The computing device 800 includes a storage device storing coordinatesof the area of interest. The computing device executes a specializedtrajectory module 802 that produces the data for the appropriate flightplan over an area of interest. The trajectory module 802 allows thedetermination of a series of trajectories. Each trajectory is determinedbased on a logarithmic spiral curve derived from a predetermined rangeof basis angles and selecting a constant tangent angle between a radialline from the location of an image sensor to a target location, and atangent line to the logarithmic spiral curve at the location of theimage sensor. The trajectory module 802 stores the series oftrajectories in the storage device. The trajectory module 802 allows theselection of a set of trajectories from the series of trajectories thatfits the area of interest. The trajectory module 802 allows scaling theselected set of trajectories to cover the area of interest andtransforming the scaled selected set of trajectories to coordinates tocover the area of the interest. A display module of the computing device800 is coupled to the trajectory module 802. The display module maydisplay the selected set of trajectories over an image of the area ofinterest.

As explained above, the trajectories may form the basis of a flight planthat may be determined by the computing device 800. The determinedflight plan is downloaded from the computing device 800 to a droneflight control unit 810. The flight plan is then translated into flightinstructions to a UAV such as a drone 820.

The drone 820 flies the determined flight plan to obtain images of atarget 822. As explained above, the target 822 may be part of a largersegment of an area of interest. The drone 820 includes a controller 830.The controller 830 receives instructions from a transceiver 832 that isin communication with the flight controller 810. The controller 830sends commands to flight controls 834 that control the altitude, speed,and direction of the drone 820 through engine speed and azimuth angle.The controller 830 also sends commands to a camera actuator 836 toposition a camera 838 during flight. As explained above, the camera 838is positioned to maintain a constant angle to the area of interest basedon the flight path.

The process for determining a flight path for imaging an area ofinterest is as follows, understanding that the intent of the process isto use the system to create non-traditional UAV flight mission plans formapping an area photographically. The computing device 800 executes thedisclosed method to create non-traditional (non-linear, non-parallel,convergent) logarithmic spiral camera trajectories that may be used toexecute flight patterns. It is to be understood, that a similar processmay be followed for a ground based or water based vehicle that isdeployed for imaging an area.

In this example, a reusable family (set) of unit-scale logarithmicspiral trajectories is created. One set is created for 0≤Theta≤π inincrements of five (5) degrees, and a second set of trajectories iscreated for 0≤Theta≤π in increments of ten (10) degrees while theconstant tangent angle Psi is varied (0<=Psi<=π/2) using increments offive (5) degrees. The trajectory construction process is as follows: Fora given value of Psi, the value of Theta is varied throughout its entirerange of values, thus creating a single trajectory; and the processes isrepeated for an incremented value of Psi. An entire library of unitscale trajectory templates can be created in a single spreadsheet andsaved as a single .csv file. The unit scale trajectory templates arereusable.

Once the trajectory templates are calculated, based on the size andshape of a specific target (point, area, or narrow corridor), a fewtrajectories needed to cover the target area are selected by taking theeccentricity of the templates into account. The selected unit-scaletrajectory templates (waypoints) are scaled to reflect the (x,y)dimensions (extremities) of the target. In this example, the system willtransform (rotate and translate) the two-dimensional (2D) scaledtrajectories (waypoints) into three-dimensional (3D) real world StatePlane (NAD83/NADV) Cartesian coordinates. Subsequently, the Cartesianstate plane coordinate waypoints are converted into WGS84(Latitude/Longitude, elevation) coordinate waypoints.

In this example, after the non-traditional logarithmic spiral waypointshave been converted into WGS84 (Latitude/Longitude, elevation)coordinates, the process is continued with the use of UAV missionplanning software such as DJIFlightPlanner to create a traditional(linear, parallel flight lines) UAV mission plan. The software may berun on the computing device 800. The flight planning process involvesfirst defining the target area boundary points, which become the basisfor the automatic creation of a traditional (equally spaced, linear,parallel) set of flight lines. At each waypoint along the flight lines,the software operator prescribes one or more actions to be executed. Theactions may include: a) ascend/descend, b) take a photograph, c) changedirection, d) orbit a point, or e) return to base. In this example, acompleted “DJI” CSV mission plan (waypoint) file is exported. A GoogleEarth kmz file of the target area is exported for visualization ofresults. A “Litchi” mission plan (waypoint) CSV file is exported.

In this example, an operator may use a spreadsheet program to open andedit the Litchi mission plan CSV file to replace all the DJI(traditional) waypoints with the logarithmic spiral (non-traditional)waypoints, without making any changes to the attributes (actions)associated with any particular waypoint record. After the editoperations are complete, the CSV file is re-saved with theupdated/edited trajectories (modified mission plan CSV file) stored inthe computing device 800.

Using the Litchi Mission Hub, the edited Litchi mission plan CSV file isopened and updated as necessary using Litchi protocols and procedures toarchive a non-traditional UAV mission plan CSV file. The mission plan isthen sent to the flight controller 810, which in this example is aLitchi flight controller. Once the plan is loaded, the drone 820 may beflown based on the non-traditional UAV mission plan trajectories.

The present disclosure relates generally to the field of digital imagedata capture (terrestrial and/or aerial photography) using various typesof cameras deployed by various types of mechanisms along a logarithmicspiral trajectory for the purpose of developing a) three-dimensionalpoint clouds and b) associated orthophotography and DEM (usingStructure-from-Motion techniques) each depicting the three dimensionaltopographical scene that was photographed. The scene may or may notinclude man-made structures in either open (outdoor) or enclosed(indoor) environments.

FIG. 9 illustrates non-traditional logarithmic spiral trajectories of aflight plan projected onto an area of interest. The trajectories aredetermined using the process described above. In this example, the image900 illustrates flight trajectories 910 displayed (using Google Earth)over the target area, which is Bessemer Airport in Bessemer, Ala. Thetrajectories extend beyond the ends of a runway 920 and well beyond theedges of the developed airport property to ensure that the resulting SfMmodel covers the entire target area.

This example, demonstrates the development of the logarithmic spiraltrajectories into an actual non-traditional manned (piloted) aircraftmission plan. The example area of interest is a commercial airport(Bessemer Airport). The site is 0.5 miles wide and 2.2 miles long or 1.1square miles in area and may be characterized as a long wide corridor.

Since a manned fixed-wing aircraft is used in this example, wider morerounded logarithmic spiral trajectories are desirable to facilitateeasier transition from one flight loop to another. Thus, trajectorieswith constant tangent angles (Psi) of 40, 50, and 60 degrees were chosenfrom the family/set of unit-scale trajectories. The unit-scaletrajectories were re-scaled (X-Scale=11,147 and Y-Scale=7,500) to ensurethe entire area of interest was covered to provide a geometricallycorrect result. The required transformation matrix is computed; and thetrajectory waypoint coordinates are transformed into a local horizontalCartesian coordinate system. The local horizontal Cartesian waypoints(Easting, Northing, elevation) are loaded into mapping software such asGlobal Mapper and projected into state plane coordinates (Alabama-Westin this example) to provide geographically correct flight trajectories.

Shape (.shp) files were exported (for both points and lines) forsubsequent use with the CartoMobile GIS mapping software running on theiPad/iPhone, providing a fundamental pilot navigation aid. A CSV (text)file may be exported (Alabama-West State Plane Coordinates) for use witha proprietary pilot navigation software package (PNAV). A Google Earth.kmz file of the area of interest was exported for visualization(quality assurance) purposes using Google Earth. The kmz file may beused to transfer the flight plan (trajectories/waypoints) to acommercially available flight planning software such as ForeFlight.

In this example, a manned aircraft was selected to fly the mission'slogarithmic spiral trajectories guided by the PNAV pilot navigationsoftware. The example aircraft is a twin-engine Partenavia aircraft usedregularly for collecting traditional aerial photography (bothorthophotography and oblique photos). The Partenavia aircraft is wellsuited for this type of mission since it is a very stable aerialphotography platform capable of fairly slow flight speeds in the rangeof 60 to 70 knots (air speed), and it has a large camera bay. ThePartenavia aircraft is equipped with a moving map pilot navigationsystem (PNAV).

FIG. 10A shows an overview image of an example PNAV display 1000 on anaircraft to assist a pilot in flying a mission plan to take images of anarea. The display 1000 illustrates an overview of the mission plan. Theindividual flight trajectories are shown as lines 1010 while eachwaypoint along the individual flight trajectories is shown as a point(dot) 1020 along with a corresponding waypoint identification number.

The display 1000 includes a distance marker 1030, a ground speed marker1032, and an altitude marker 1034. The distance marker 1030 at the topof the display 1000 indicates the aircraft's distance (left/right) ofthe current flight line. The ground speed marker 1032 on the left sideof the display 1000 indicates the current ground speed in knots. Thealtitude marker 1034 at the right side of the display 1000 indicates thecurrent height-above-ground in feet. The display 1000 includes a dataitems array 1036 at the bottom that provides the pilot with importantnavigation and flight data and offers controls for the PNAV display 1000and data recording capabilities.

FIG. 10B shows the PNAV display 1000 in FIG. 10A shortly after enteringthe flight pattern determined from the selected trajectories. Thelocation of the aircraft is represented by a symbol PL 1050. The datadisplayed at the bottom of the screen 1000 shows that the aircraft is0.30 miles from (Dist to Start) the start point; is approaching waypointnumber 228 (distance of 0.12 miles) shown as point 1052 at a heading of45.3 degrees and an arrival time of 0.07 minutes; while the anticipated(Next+1) waypoint is numbered 229 shown as point 1054 at a heading of48.7 degrees. The aircraft heading is constantly changing as theaircraft proceeds from one waypoint to the next.

FIG. 10C shows the PNAV display 1000 in FIG. 10A with the aircraftapproximately half way through its first trajectory. The data displayedat the bottom of the screen 1000 shows the aircraft is 1.08 miles from(Dist to Start) the start point; is approaching waypoint number 175(distance of 0.22 miles) shown as point 1060 at a heading of 48.4degrees and an arrival time of 0.13 minutes; while the anticipated(Next+1) waypoint shown as point 1062 is numbered 174 at a heading of51.8 degrees. The aircraft heading is constantly changing as it proceedsfrom one waypoint to the next.

FIG. 10D shows the PNAV display 1000 in FIG. 10A with the aircraftapproaching the end of its first trajectory with all flight trajectoriesconverging on each other. The data displayed at the bottom of the screen1000 shows that the aircraft is 1.66 miles from (Dist to Start) thestart point; is approaching waypoint number 173 (distance of 0.13 miles)shown as a point 1070 at a heading of 55.1 degrees and an arrival timeof 0.08 minutes; while the anticipated (Next+1) waypoint is numbered 172at a heading of 58.6 degrees shown as a point 1072. The aircraft headingis constantly changing as it proceeds from one waypoint to the next.

As the end (convergent) point of the trajectories is approached, thepilot will either choose to follow the waypoints through and along thesame trajectory loop or break away, make a smooth transition turn, andpick up his next trajectory to return to the “start” end of the targetarea. It is recommended that the pilot plan (and perhaps rehearse) hisanticipated maneuvers ahead of the time of the flight. The PNAV pilotnavigation system does have a “simulation” mode to support such pilotplanning efforts and/or rehearsals.

FIG. 11 is a flow diagram of the software instructions executed by thetrajectory module of the system 800 in FIG. 8 to produce a flight plan.The flow diagram in FIG. 11 is representative of example machinereadable instructions for the computing device 800 in FIG. 8 . In thisexample, the machine readable instructions comprise an algorithm forexecution by: (a) a processor; (b) a controller; and/or (c) one or moreother suitable processing device(s). The algorithm may be embodied insoftware stored on tangible media such as, for example, a flash memory,a CD-ROM, a floppy disk, a hard drive, a digital video (versatile) disk(DVD), or other memory devices. However, persons of ordinary skill inthe art will readily appreciate that the entire algorithm and/or partsthereof can alternatively be executed by a device other than a processorand/or embodied in firmware or dedicated hardware in a well-known manner(e.g., it may be implemented by an application specific integratedcircuit [ASIC], a programmable logic device [PLD], a field programmablelogic device [FPLD], a field programmable gate array [FPGA], discretelogic, etc.). For example, any or all of the components of theinterfaces can be implemented by software, hardware, and/or firmware.Also, some or all of the machine readable instructions represented bythe flowchart of FIG. 11 may be implemented manually. Further, althoughthe example algorithm is described with reference to the flowchartillustrated in FIG. 11 , persons of ordinary skill in the art willreadily appreciate that many other methods of implementing the examplemachine readable instructions may alternatively be used. For example,the order of execution of the blocks may be changed, and/or some of theblocks described may be changed, eliminated, or combined.

The computing device 800 first calculates a series of logarithmic curves(1100). A series of constant tangent angles (Psi) are selected, forexample 18 angles at five degree intervals from zero to 90 degrees. Foreach of the constant tangent angles, a range of basis angles (Theta) areselected, for example 37 angles at five degree intervals from zero to180 degrees. Thus, a series of logarithmic curves, each having aconstant tangent angle and a constant predetermined angle, which resultin 72 curves in this example.

The computing device 800 then rotates and mirrors the logarithmic curvesto produce corresponding trajectory templates (1102). The resultingtrajectory templates are stored as a library of trajectory templates(1104). The operator of the computing device 800 then determines thetarget area by viewing an image file (e.g., Google Earth) or a map file(e.g., CartoMobile) or coordinate file from a GIS system (1106). Anappropriate scale factor is selected to fit the trajectories to thetarget area (1108). The trajectories in the library are then scaledbased on the selected scaled factor (1110).

Appropriate trajectory templates are then selected from the library oftrajectory templates to cover the target area (1112). Each of theselected trajectory waypoints are then converted to coordinates relatingto the target area (1116). The coordinates of the trajectories and thewaypoints are converted to flight data in the form of data files such as.csv and .kml files. The flight data is loaded into an appropriateflight controller in the case of an unmanned aircraft or an autopilotsystem in the case of a manned aircraft (1118).

FIG. 12 illustrates an example computing system 1200, in which thecomponents of the computing system are in electrical communication witheach other using a bus 1202. The system 1200 includes a processing unit(CPU or processor) 1230, and a system bus 1202 that couples varioussystem components, including the system memory 1204 (e.g., read onlymemory [ROM] 1206 and random access memory [RAM] 608), to the processor1230. The system 1200 can include a cache of high-speed memory connecteddirectly with, in close proximity to, or integrated as part of theprocessor 1230. The system 1200 can copy data from the memory 1204and/or the storage device 1212 to the cache 1228 for quick access by theprocessor 1230. In this way, the cache can provide a performance boostfor processor 1230 while waiting for data. These and other modules cancontrol or be configured to control the processor 1230 to performvarious actions. Other system memory 1204 may be available for use aswell. The memory 1204 can include multiple different types of memorywith different performance characteristics. The processor 1230 caninclude any general purpose processor and a hardware module or softwaremodule, such as module 1 1214, module 2 1216, and module 3 1218 embeddedin storage device 1212. The hardware module or software module isconfigured to control the processor 1230, as well as a special-purposeprocessor where software instructions are incorporated into the actualprocessor design. The processor 1230 may essentially be a completelyself-contained computing system, containing multiple cores orprocessors, a bus, memory controller, cache, etc. A multi-core processormay be symmetric or asymmetric.

To enable user interaction with the computing device 1200, an inputdevice 1220 is provided as an input mechanism. The input device 1220 cancomprise a microphone for speech, a touch-sensitive screen for gestureor graphical input, keyboard, mouse, motion input, and so forth. In someinstances, multimodal systems can enable a user to provide multipletypes of input to communicate with the system 1200. In this example, anoutput device 1222 is also provided. The communications interface 1224can govern and manage the user input and system output.

Storage device 1212 can be a non-volatile memory to store data that areaccessible by a computer. The storage device 1212 can be magneticcassettes, flash memory cards, solid state memory devices, digitalversatile disks, cartridges, random access memories (RAMs) 1208, readonly memory (ROM) 1206, and hybrids thereof.

The controller 1210 can be a specialized microcontroller or processor onthe system 1200, such as a BMC (baseboard management controller). Insome cases, the controller 1210 can be part of an Intelligent PlatformManagement Interface (IPMI). Moreover, in some cases, the controller1210 can be embedded on a motherboard or main circuit board of thesystem 1200. The controller 1210 can manage the interface between systemmanagement software and platform hardware. The controller 1210 can alsocommunicate with various system devices and components (internal and/orexternal), such as controllers or peripheral components, as furtherdescribed below.

The controller 1210 can generate specific responses to notifications,alerts, and/or events, and communicate with remote devices or components(e.g., electronic mail message, network message, etc.) to generate aninstruction or command for automatic hardware recovery procedures, etc.An administrator can also remotely communicate with the controller 610to initiate or conduct specific hardware recovery procedures oroperations, as further described below.

The controller 1210 can also include a system event log controllerand/or storage for managing and maintaining events, alerts, andnotifications received by the controller 610. For example, thecontroller 1210 or a system event log controller can receive alerts ornotifications from one or more devices and components, and maintain thealerts or notifications in a system event log storage component.

Flash memory 1232 can be an electronic non-volatile computer storagemedium or chip that can be used by the system 1200 for storage and/ordata transfer. The flash memory 1232 can be electrically erased and/orreprogrammed. Flash memory 1232 can include EPROM (erasable programmableread-only memory), EEPROM (electrically erasable programmable read-onlymemory), ROM, NVRAM, or CMOS (complementary metal-oxide semiconductor),for example. The flash memory 1232 can store the firmware 1234 executedby the system 1200 when the system 1200 is first powered on, along witha set of configurations specified for the firmware 1234. The flashmemory 1232 can also store configurations used by the firmware 1234.

The firmware 1234 can include a Basic Input/Output System orequivalents, such as an EFI (Extensible Firmware Interface) or UEFI(Unified Extensible Firmware Interface). The firmware 1234 can be loadedand executed as a sequence program each time the system 1200 is started.The firmware 1234 can recognize, initialize, and test hardware presentin the system 1200 based on the set of configurations. The firmware 1234can perform a self-test, such as a POST (Power-on-Self-Test), on thesystem 1200. This self-test can test the functionality of varioushardware components such as hard disk drives, optical reading devices,cooling devices, memory modules, expansion cards, and the like. Thefirmware 1234 can address and allocate an area in the memory 1204, ROM1206, RAM 1208, and/or storage device 1212, to store an operating system(OS). The firmware 1234 can load a boot loader and/or OS, and givecontrol of the system 1200 to the OS.

The firmware 1234 of the system 1200 can include a firmwareconfiguration that defines how the firmware 1234 controls varioushardware components in the system 1200. The firmware configuration candetermine the order in which the various hardware components in thesystem 1200 are started. The firmware 1234 can provide an interface,such as an UEFI, that allows a variety of different parameters to beset, which can be different from parameters in a firmware defaultconfiguration. For example, a user (e.g., an administrator) can use thefirmware 1234 to specify clock and bus speeds; define what peripheralsare attached to the system 1200; set monitoring of health (e.g., fanspeeds and CPU temperature limits); and/or provide a variety of otherparameters that affect overall performance and power usage of the system1200. While firmware 1234 is illustrated as being stored in the flashmemory 1232, one of ordinary skill in the art will readily recognizethat the firmware 1234 can be stored in other memory components, such asmemory 1204 or ROM 1206.

System 1200 can include one or more sensors 1226. The one or moresensors 1226 can include, for example, one or more temperature sensors,thermal sensors, oxygen sensors, chemical sensors, noise sensors, heatsensors, current sensors, voltage detectors, air flow sensors, flowsensors, infrared thermometers, heat flux sensors, thermometers,pyrometers, etc. The one or more sensors 1226 can communicate with theprocessor, cache 1228, flash memory 1232, communications interface 1224,memory 1204, ROM 1206, RAM 1208, controller 1210, and storage device1212, via the bus 1202, for example. The one or more sensors 1226 canalso communicate with other components in the system via one or moredifferent means, such as inter-integrated circuit (I2C), general purposeoutput (GPO), and the like. Different types of sensors (e.g., sensors1226) on the system 1200 can also report to the controller 1210 onparameters, such as cooling fan speeds, power status, operating system(OS) status, hardware status, and so forth. A display 1236 may be usedby the 1200 to provide graphics related to the applications that areexecuted by the controller 1210, or the processor 1230.

FIG. 13 illustrates an example computer system 1300 having a chipsetarchitecture that can be used in executing the described method(s) oroperations, and generating and displaying a graphical user interface(GUI). Computer system 1300 can include computer hardware, software, andfirmware that can be used to implement the disclosed technology. System1300 can include a processor 1310, representative of a variety ofphysically and/or logically distinct resources capable of executingsoftware, firmware, and hardware configured to perform identifiedcomputations. Processor 1310 can communicate with a chipset 1302 thatcan control input to and output from processor 1310. In this example,chipset 1302 outputs information to output device 1314, such as adisplay, and can read and write information to storage device 1316. Thestorage device 1316 can include magnetic media, and solid state media,for example. Chipset 1302 can also read data from and write data to RAM1318. A bridge 1304 for interfacing with a variety of user interfacecomponents 1306, can be provided for interfacing with chipset 1302. Userinterface components 1306 can include a keyboard, a microphone, touchdetection and processing circuitry, and a pointing device, such as amouse.

Chipset 1302 can also interface with one or more communicationinterfaces 1308 that can have different physical interfaces. Suchcommunication interfaces can include interfaces for wired and wirelesslocal area networks, for broadband wireless networks, and for personalarea networks. Further, the machine can receive inputs from a user viauser interface components 1306, and execute appropriate functions, suchas browsing functions by interpreting these inputs using processor 1310.

Moreover, chipset 1302 can also communicate with firmware 1312, whichcan be executed by the computer system 1300 when powering on. Thefirmware 1312 can recognize, initialize, and test hardware present inthe computer system 1300 based on a set of firmware configurations. Thefirmware 1312 can perform a self-test, such as a POST, on the system1300. The self-test can test the functionality of the various hardwarecomponents 1302-1318. The firmware 1312 can address and allocate an areain the RAM memory 1318 to store an OS. The firmware 1312 can load a bootloader and/or OS, and give control of the system 1300 to the OS. In somecases, the firmware 1312 can communicate with the hardware components1302-1310 and 1314-1318. Here, the firmware 1312 can communicate withthe hardware components 1302-1310 and 1314-1318 through the chipset1302, and/or through one or more other components. In some cases, thefirmware 1312 can communicate directly with the hardware components1302-1310 and 1314-1318.

It can be appreciated that example systems 1200 and 1300 can have morethan one processor (e.g., 1230, 1310), or be part of a group or clusterof computing devices networked together to provide greater processingcapability.

As used in this application, the terms “component,” “module,” “system,”or the like, generally refer to a computer-related entity, eitherhardware (e.g., a circuit), a combination of hardware and software,software, or an entity related to an operational machine with one ormore specific functionalities. For example, a component may be, but isnot limited to being, a process running on a processor (e.g., digitalsignal processor), a processor, an object, an executable, a thread ofexecution, a program, and/or a computer. By way of illustration, both anapplication running on a controller, as well as the controller, can be acomponent. One or more components may reside within a process and/orthread of execution and a component may be localized on one computerand/or distributed between two or more computers. Further, a “device”can come in the form of specially designed hardware; generalizedhardware made specialized by the execution of software thereon thatenables the hardware to perform specific function; software stored on acomputer-readable medium; or a combination thereof.

The terminology used herein is for the purpose of describing particularembodiments only, and is not intended to be limiting of the invention.As used herein, the singular forms “a,” “an,” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. Furthermore, to the extent that the terms “including,”“includes,” “having,” “has,” “with,” or variants thereof, are used ineither the detailed description and/or the claims, such terms areintended to be inclusive in a manner similar to the term “comprising.”

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art. Furthermore, terms, such as those definedin commonly used dictionaries, should be interpreted as having a meaningthat is consistent with their meaning in the context of the relevantart, and will not be interpreted in an idealized or overly formal senseunless expressly so defined herein.

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. Numerous changes to the disclosedembodiments can be made in accordance with the disclosure herein,without departing from the spirit or scope of the invention. Thus, thebreadth and scope of the present invention should not be limited by anyof the above described embodiments. Rather, the scope of the inventionshould be defined in accordance with the following claims and theirequivalents.

Although the invention has been illustrated and described with respectto one or more implementations, equivalent alterations and modificationswill occur or be known to others skilled in the art upon the reading andunderstanding of this specification and the annexed drawings. Inaddition, while a particular feature of the invention may have beendisclosed with respect to only one of several implementations, suchfeature may be combined with one or more other features of the otherimplementations as may be desired and advantageous for any given orparticular application.

What is claimed is:
 1. A method of developing a flight path over ageographic area, corridor, or point of interest, the method comprising:determining a set of trajectories, each trajectory determined based on alogarithmic spiral curve derived from a basis angle from a predeterminedrange of basis angles and selecting a constant tangent angle between aradial line from the location of an aircraft to a target location, and atangent line to the slope of the logarithmic spiral curve at thelocation of the aircraft on the logarithmic spiral curve; scaling theset of trajectories to cover the geographic area, corridor, or point ofinterest; transforming the set of trajectories to coordinatescorresponding to the geographic area, corridor, or point of interest;and storing a flight path based on the set of trajectories in a storagedevice.
 2. The method of claim 1, further comprising combining the setof scaled trajectories as a closed loop flight path over the geographicarea, corridor, or point of interest.
 3. The method of claim 2, whereinthe set of trajectories are selected from a series of trajectories, andwherein the trajectories of the series of trajectories are based onincremented values between each constant tangent angle.
 4. The method ofclaim 3, wherein the predetermined range of basis angles are incrementedby a selected angle in the range.
 5. The method of claim 2, wherein theclosed loop flight plan includes a set of sequenced waypoints determinedby translating and rotating trajectory waypoints based on the geographicarea, corridor, or point of interest.
 6. The method of claim 5, whereinthe waypoints are input in a flight controller.
 7. The method of claim6, wherein the aircraft is a drone.
 8. The method of claim 5, whereinthe waypoints are displayed on a PNAV navigation display in an aircraftcockpit.
 9. The method of claim 1, wherein the logarithmic curve isdetermined by the equation:r=+e{circumflex over ( )}((Theta−π)*cot(Psi)) wherein r is the distancefrom the origin to the logarithmic curve, Theta is the predeterminedbasis angle, and Psi is the constant tangent angle.
 10. The method ofclaim 1, wherein the logarithmic curve is determined by one of aFibonacci series, mathematical series, a series of Bezier curves, or aseries of B-splines.
 11. A system to determine a flight path for flyingover a geographic area, corridor, or point of interest, the systemcomprising: a storage device storing coordinates of the geographic area,corridor, or point of interest; a trajectory module coupled to thestorage device, the trajectory module operative to: determine a set oftrajectories, each trajectory determined based on a logarithmic spiralcurve derived from a basis angle from a predetermined range of basisangles and selecting a constant tangent angle between a radial line fromthe location of an aircraft to a target location, and a tangent line tothe slope of the logarithmic spiral curve at the location of theaircraft on the logarithmic spiral curve; store the set of trajectoriesin the storage device, scale the set of trajectories to cover thegeographic area, corridor, or point of interest, and transform thescaled set of trajectories to coordinates to cover the geographic area,corridor, or point of interest; and a display module coupled to thetrajectory module, the display module displaying the scaled set oftrajectories over the geographic area, corridor, or point of interest.12. The system of claim 11, wherein the set of trajectories are createdfrom a series of trajectories based on incremented values between eachconstant tangent angle.
 13. The system of claim 11, wherein thepredetermined range of basis angles are incremented by a selected anglein the range.
 14. The system of claim 11, further comprising a flightplan module operable for creating a flight plan for an aircraft based onthe set of scaled trajectories.
 15. The system of claim 14, wherein theflight plan is a closed loop flight path determined from combining thescaled set of trajectories.
 16. The system of claim 14, wherein theflight plan includes a set of sequenced waypoints determined bytranslating and rotating trajectory waypoints based on the geographicarea, corridor, or point of interest.
 17. The system of claim 16,wherein the waypoints are in a format for input in a flight controller.18. The system of claim 14, wherein the aircraft is a drone.
 19. Thesystem of claim 14, wherein the aircraft is piloted, and wherein thewaypoints are displayed on a PNAV navigation display in an aircraftcockpit.
 20. The system of claim 11, wherein the logarithmic curve isdetermined by the equation:r=+e{circumflex over ( )}((Theta−π)*cot(Psi)) wherein r is the distancefrom the origin to the logarithmic curve, Theta is the predeterminedbasis angle, and Psi is the constant tangent angle.
 21. The system ofclaim 11, wherein the logarithmic curve is determined by one of aFibonacci series, mathematical series, a series of Bezier curves, or aseries of B-splines.
 22. A vehicle control system for determining aroute over a geographic area, corridor, or point of interest, the systemcomprising: a storage device storing a route plan, the route plan basedon determining a set of trajectories, each trajectory determined basedon a logarithmic spiral curve derived from a basis angle from apredetermined range of basis angles and selecting a constant tangentangle between a radial line from the location of a vehicle to a targetlocation, and a tangent line to the slope of the logarithmic spiralcurve at the location of the vehicle on the logarithmic spiral curve;scaling the set of trajectories to cover the geographic area, corridor,or point of interest; transforming the set of trajectories tocoordinates corresponding to the geographic area, corridor, or point ofinterest; and generating a set of waypoints for the route plan; and adisplay displaying the route plan.
 23. The vehicle control system ofclaim 22, further comprising a controller coupled to the storage deviceand controls of the vehicle, the controller operating the controls ofthe vehicle according to the route plan.